EL YO CONSCIENTE

The correlation coefficient and cross-correlation functions provide tools to analyze the aggregate relationship between the large- and small-scale signals in both temporal and physical space. How- ever, much of what is known about the large-scale motions in the wall-bounded flows can be better described in spectral space, where the size of the motions, in some statistical sense, can be character- ized by their energetic contributions at particular frequencies or wavelengths. Normally, this spectral analysis is performed on the instantaneous velocity signals themselves; however, by performing a spectral analysis on the filtered signals, a new perspective on the modulation effect can be gained, and in particular the influence of the enveloping procedure since it is the envelope procedure which ultimately attributes low frequency content to theuR signal to produceuS.

10−1 100 101 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 τ U/δ y/ δ R max

Figure 7.7: The zero-crossing locations from the correlation coefficients (squares) and theπ/2 phase- change locations from the cross-correlations (circles), with a best-fit (blue) assigned to the mean value, y/δ ≈ 10−0.8(_{τ U}/_δ)−0.2_{. Also, the maximum positive value of the correlation coefficient,}

with a best-fit (red) of Rmax ≈ 10−0.5(τ U/δ)−0.1. Values are recorded for the most downstream

measurement, withReθ≈4100.

By transforming the cross-correlation, r(∆t), from the temporal domain into the frequency domain by Fourier transform, the cospectral density,rc(uL, uS)between the large-scale fluctuations

and the envelope of the small-scales is produced. The expression representing this quantity is simply equation 7.2 without the final inverse Fourier transform,F−1_{, shown as equation 7.3.}

rc(uL, uS) = F (uL− ⟨uL⟩) F (uS− ⟨uS⟩) ∗

(7.3) The transform process, as above, involves windowing the time series and averaging across the windows. However, an additional subtlety arises in the frequency domain about the form of the aver- aging. In order to smooth the power spectral density estimate, the PSD of each window is averaged ‘incoherently’ in the frequency domain, following Welch’s method; alternatively, the windows can be averaged ‘coherently’ in the time domain, which under certain conditions (of stationarity and ergod- icity) can preserve phase information. Assuming the long time series fulfill, at least roughly, these conditions, the coherent approach was employed in the hope of preserving the relevant phase infor- mation between different frequencies. Therefore, the argument of the cospectrum, arg[rc(uL, uS)]

represents the phase difference between each discrete Fourier component of the large scales with the envelope of small-scale motions. The normalized cospectral power can be defined such that the integral of the power in the cospectral density is equal to the covariance of the large- and small-scale signals, as in equation 7.4

Co(uL, uS) =

rc(uL, uS)rc(uL, uS)∗

cov(uL, uS)

(7.4) Calculating the cospectral density at a single wall-normal location, shown in figure 7.8, provides insight into the general shape of the function. In order to represent the key features of the smoothed spectrum, a fitting technique was employed, whereby a modified Gaussian function is fitted to the underlying spectral data, as described in appendix 7.8.2. Then, by calculating the cospectral density (and its fitted analogue) at each wall-normal location, a map of the cospectral density can be produced, shown in figure 7.9, where a dashed line is used to indicate local-velocity-dependent size of the filter cutoff used to separate the large- and small-scale motions.

The cospectral density map in figure 7.9 shows a peak across most of the wall-normal locations at f δ/U∞ ≈ 0.1, which indicates the dominant large-scale motion among all of the large scales

participating in the phase relationship with the small-scale envelope. Moreover, the fitted version of the map (figure 7.9b, where the modified Gaussian fit was used to provide a smoothed version of the cospectrum) highlights two distinct regions of dominant large scales: one within the buffer layer, and one in the outer region of the boundary layer.

Guala et al. [2011] also observed two regions in which large-scale motions showed a discernible effect on small scales, by applying conditional averaging to the velocity signals based on the sign of large-scale motions, and then examining the difference between the standard premultiplied power spectral densities of the conditionally averaged signals (positive large-scale spectra minus negative) in the atmospheric surface layer. They noted two peaks in power difference, one near the wall in the buffer layer, situated at higher frequencies, and one further from the wall, at lower frequencies, showing that greater small-scale intensity is associated with positive large-scale excursions near the wall. Importantly, it is also apparent from their results that a variety of large scales, both near the wall and farther away, are involved in this interaction.

In the cospectral density map, it appears that the lower frequency peak is actually situated nearer to the wall, although only slightly so, meaning that near the wall, the increased small-scale activity is associated with the larger wavelength range of large-scale motions, and farther from the wall, the increased small-scale activity is associated with relatively ‘smaller’ large scales. Like the results of Guala et al. [2011], this indicates that different regions of the boundary layer have different phase relationships between large- and small-scale motions, although here the evidence is more specific that near the wall there is a broader range of larger scales that appear to have a footprint on the small-scale envelope.

The trace of the peak in the cospectral density across wall-normal locations appears to yield a distinctive shape over the range of Reynolds number available on the flat plate. The location of this ridgeline represents what will be referred to as the ‘dominant interacting scale’,λx, involved in the

10−3 10−2 10−1 100 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 f δ/U ∞ Co(f)

Figure 7.8: The normalized cospectral density,Co(f δ/U∞), aty/δ≈0.17, in black. The red line is

the best fit to the extended Gaussian function, defined in equation 7.14. The cospectral density is displayed in a normalized but not premultiplied form in order to identify the frequencies at which the interaction between large- and small-scales is dominant, but not necessarily energetically so, because the energetic strength does not indicate phase interaction strength, as can be observed by analysis of two amplitude-modulating sinusoids.

f δ/U ∞ y/ δ 10−3 10−2 10−1 100 10−2 10−1 100 Co(f) × 10−2 0.25 0.5 0.75 1 f δ/U ∞ y/ δ 10−3 10−2 10−1 100 10−2 10−1 100 Co(f) × 10−2 0.25 0.50 0.75 1

Figure 7.9: (Left) The cospectral density for the cross-correlation ofuL anduS defined by temporal

means from the hotwire measurements. The peaks from the amplitude at each wall-normal location are denoted by circles. The filter size of τ = 1 δ/U is marked by a dashed line, which varies as a function of convective velocity. (Right) The same spectral map, constructed from the best-fit extended Gaussian functions, in order to identify the peaks, which are traced by the solid black line. The blue dotted line corresponds to the power law for the VLSM reported by Monty et al. [2009], translated to the frequency domain via Taylor’s hypothesis using the present mean velocity profile.

phase relationship with the small-scale envelope. The dominant interacting scale can be expressed as a simple power law function of Reynolds number and wall-normal location, as in equation 7.5, where both frequency and wavelength are employed for subsequent convenience. The robust fitting process used to obtain the power law relations is described in detail in appendix 7.8.2. The dominant interacting scale is that scale, among all of the large scales in the flow, which is most strongly correlated with the envelope of small-scale motions.

f δ/U∞=10 −0.38( Reθ)−0.26(y/δ)−0.15 λx/δ=100.32(Reθ)0.28(y/δ)0.34 ≈20(y/δ)0.34 f δ/U∞=10 −0.23(_Re τ)−0.36(y/δ)−0.15 λx/δ=100.15(Reτ)0.38(y/δ)0.34 ≈20(y/δ)0.34 (7.5)

The scaling of the dominant interacting scale involved in the phase relationship can be interpreted in the context of the other key structural features of wall-bounded flows. Monty et al. [2009] reviewed a number of these key features, as observed through the composite spectral maps, including: the highly energetic peak near the wall aty+≈_{15, λ}+

x≈1000, a dominant (LSM) peak fory/δ>0.3, λx/δ≈

2–3, and also a secondary (VLSM) peak aty/δ≈0.06, λx/δ≈6. The secondary peak tends to persist

beyondy/δ>0.3 along with the LSM peak for internal flows, but tends to shift to lower wavelengths for the boundary layer such that only the LSM peak is observed in the outer region. The size of the LSM peak, withλx/δ≈2–3, is consistent with the typical size of the intermittent bulges in the edge

of the boundary layer, as identified by Kovasznay et al. [1970] and Falco [1977]. The implication is that the intermittency at the edge of external flows tends to enforce its dominant scale on the outer region of flow, to the exclusion of a clear VLSM signature. However, the scale of the dominant interacting scale from the cospectral density appears to be quite similar to the size of the VLSM in internal flows. To make a more careful comparison between the two structures, an expansion of the dependence of the dominant interacting scale on wall-normal location can be compared with that of the VLSM. Monty et al. [2009] report that VLSM scaling in the internal flows tends to follow a power law of the formλEx/δ≈23(y/δ)0.43, similar to the scaling inferred from the boundary layer data of

Kim and Adrian [1999] which fits (exactly, with only two points provided)λEx/δ≈20(y/δ)0.38. Both

of these fits for the VLSM scaling agree strongly with the fit for the dominant interacting scale identified above, both in exponent and intercept, showing not only a similar physical size but also gradient across the boundary layer.

By overlaying the peaks from the cospectral density on the premultiplied spectrum of the stream- wise turbulence, from Jacobi and McKeon [2011a], this similarity can be observed quite clearly. Figure 7.10 shows this overlay for a raw spectral map. In some sense, the dominant interacting scale appears precisely where the VLSM would have, if not for the dominance of the boundary layer inter- mittency. Therefore, the presence of the modulating scale, as inferred from the cospectral density, indicates that VLSM play an important part even in external flows, even if their spectral signature

y/ δ f δ/U ∞ 10−3 10−2 10−1 100 10−2 10−1 100 f φ(f)/u2 τ 0 0.5 1 >1.5

Figure 7.10: The premultiplied spectral map of the streamwise turbulent fluctuations from the hotwire measurements, again at the furthest downstream measurement location, corresponding to

Reθ ≈ 4100, in order to highlight the burgeoning double peak. The peaks from the amplitude at

each wall-normal location are denoted by × symbols; the ridgeline from the cospectral density is marked in circles with a solid line for the best fit, following the notation in figure 7.9. The dashed line corresponds to the power law for the VLSM reported by Monty et al. [2009], translated to the frequency domain via Taylor’s hypothesis using the present mean velocity profile; the dotted line represents the LSM atf δ/U∞≈1/3.

not apparent by standard means of analysis.

In addition to identifying the location of the dominant large-scale contribution to the phase relationship with the small-scale envelope, the cospectral density can also identify the precise phase difference between that dominant contribution and the envelope. This phase information for the cumulative effect of all scales was inferred earlier from the cross-correlation maps, in section 7.4. With the cospectrum, we are able to isolate the effect of a single mode of the large-scale motions, and therefore it is convenient to reiterate physically how this phase information can manifest itself in the physical boundary layer. Figure 7.11 shows two sample phase profiles on the left, where the phase profile is recorded from the phase at a particular frequency component of the cospectrum across the boundary layer and represents the phase difference between that particular frequency of the large scale and the envelope of small scales, as described above. On the right side of figure 7.11 are sketches of the corresponding shapes of the large-scale mode (with a fixed downstream inclination angle) and the small-scale envelope (with an inclination set by the relative phase with the large-scale mode). The case of the small-scale envelope leading the large-scale mode, corresponding to a positive phase difference, is shown on top. The negative case is shown on bottom. It should be emphasized that the negative case allows for two possible interpretations: a net lag, or a phase lead by more than half a period. The choice between these two interpretations is made by the context of the rest of the

phase profile, to maintain continuity of phase, as will be discussed in the perturbed flow, in section 8.5.

Returning to the measured cospectrum, figure 7.12 shows both a map of the phase differences over the full range of frequency components, as well as the phase difference following along the ridgeline of the dominant frequencies. Despite the caveats about the use of coherent averaging of the spectral windows, the resulting phase map shows a reasonable representation of the phase shift across the boundary layer, entirely consistent with the cross-correlation approach for the aggregation of the large scales, shown in figure 7.2. In particular, both techniques show that the phase appears to hover about π/2 for a broad range of wall-normal locations about the zero-crossing location of the correlation coefficient in the region of logarithmic layer.

By employing the cospectral density maps, it was possible to identify a dominant interacting scale on the order of the size of VLSM and to show that this dominant scale also tends to scale, across the boundary layer, following a power law similar to that governing VLSM scaling in internal flows, which strongly indicates the dynamical significance of the VLSM even far from the wall in an external flow. Moreover, the coherently averaged cospectral density revealed that this dominant interacting scale expressed precisely the phase relationship observed in the cross-correlation analysis of the aggregate large-scale signal.